Surface finite element approximation of spherical Whittle--Matérn Gaussian random fields

Abstract

Spherical Matérn-Whittle Gaussian random fields are considered as solutions to fractional elliptic stochastic partial differential equations on the sphere. Approximation is done with surface finite elements. While the non-fractional part of the operator is solved by a recursive scheme, a quadrature of the Dunford-Taylor integral representation is employed for the fractional part. Strong error analysis is performed, obtaining polynomial convergence in the white noise approximation, exponential convergence in the quadrature, and quadratic convergence in the mesh width of the discretization of the sphere. Numerical experiments for different choices of parameters confirm the theoretical findings.

Galerkin--Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds

“…for some parameters κ > 0, β > 1/2. Note in particular, that the conditionβ > 1/2 is in accordance with Condition (34) in Theorem 5.4 and that the parameter β is directly linked to the smoothness of the resulting field, which is ν − 1 times differentiable in the mean-square sense, when taking ν = (2β − 1) (Jansson et al, 2021). As for the parameter κ, it can be interpreted as a scaling parameter.…”

“…On top of the standard Galerkin error, this approach exhibits an additional geometric consistency error due to the fact that the manifold is approximated by a polyhedral surface. Even though we did not explicitly study this additional error, evidence suggest that Assumption 5.1, which is used to derive the convergence rates, still holds in this case: indeed, Bonito et al (2018) derived error estimates on the approximation of eigenvalues and eigenvectors using SFEM and the convergence analysis of Jansson et al (2021) shows convergence orders similar to those we derived. We also apply in this numerical experiment the mass lumping approximation described in Section 4.3.2.…”

“…As such, Z can be seen as an instance of a Whittle-Matérn random field on a manifold, as popularized by Lindgren et al (2011) for compact Riemannian manifolds. This class of random fields were studied by Jansson et al (2021) for the particular case where the manifold is a sphere, and for compact Riemannian manifolds by Herrmann et al (2020) and Harbrecht et al (2021).…”